Its low temperature paraelectric phase makes SrTiO\(_3\)  stand out among the other compounds of the ABO\(_3\)  perovskites family such as BaTiO\(_3\)  and PbTiO\(_3\)  [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]} where, with decreasing temperature, the polar TO phonon softens until it turns unstable and the material becomes ferroelectric.
This phase is characterised by a double well shape of the potential energy landscape along the coordinate of the so-called ferroelectric soft (FES) mode, such that at zero temperature the ground state is in a degenerate superposition of positively and negatively polarised states, which endows the material with an internal macroscopic polarization when the degeneracy is lifted by spontaneous symmetry breaking.
While the FES mode of SrTiO\(_3\)  displays a similar characteristic softening, it stabilizes at low temperatures and no ferroelectricity is observed.
This low temperature behaviour has been rationalized in terms of quantum fluctuations that prevent the formation of a macroscopic dipole [10]}.
This phenomenon of quantum paraelectricity and has been invoked to explain the behavior of other complex oxides [10]}, [12]}, [13]}, [14]}.
A number of models have been developed to describe the dielectric properties of this phase, in particular the Barrett and Vendik models have been widely used to understand quantum paraelectric behavior [15]}, [10]}, [12]}, [18]}, [19]}, [20]}.
The decisive role played by quantum fluctuations for the temperature dependent competition between the ferro- and paraelectric phases has been confirmed by quantum Monte Carlo calculations with an effective Hamiltonian for phenomenologically strained SrTiO\(_3\)  [21]}, [22]} as well as for other materials [13]}, [14]}.
The quantum paraelectric phase is therefore now widely accepted as the explanation for the low temperature behaviour of SrTiO\(_3\)  [10]}, [26]}, [27]}, however the ground state of the quantum paraelectric phase and the frequency of the FES mode at low temperature have not yet been described by a microscopic theory.
